Methods for machine learning through Gaussian processes

Abstract

A computer-implemented method for determining emission values ​​of a motor vehicle at least partially powered by an internal combustion engine during real-world driving (Real Driving Emissions (RDE)), wherein trajectories characterizing the driving cycles of the motor vehicle are provided, and wherein a Gaussian process (GP) is provided which determines the emissions of the motor vehicle when driving through the respective trajectory supplied to it, depending on the trajectories supplied to it, and wherein the respective emission values ​​of the motor vehicle when driving through these supplied trajectories are determined by means of this GP, and wherein the emission values ​​in real-world driving are determined depending on these determined emission values, characterized in that the Training the GP from a GP parameterization is performed by optimizing a PAC Bayes bound (150) using a training dataset, a prior GP family, and an accuracy target.

Description

[0001] The invention relates to a method for data modeling using Gaussian processes, a system trained with this method, a computer program comprising instructions configured to execute the method when executed on a computer, a machine-readable storage medium on which the computer program is stored, and a computer configured to execute the method. State of the art

[0002] As is known, for example, from M. Seeger, “PAC-Bayesian Generalization Error Bounds for Gaussian Process Classification”, Journal of Machine Learning Research 3, 233-269 (2002), Gaussian processes (GPs) can be used for modeling in supervised learning. They can handle large datasets, but their use in safety-critical applications is not optimal because good performance cannot be guaranteed. GPs are particularly useful due to their non-parametric nature, and although GPs, as probabilistic models, have an intrinsic degree of uncertainty, this uncertainty does not necessarily allow conclusions to be drawn about the performance, e.g., modeling goodness, of GPs with previously unprocessed data.

[0003] For example, when fitting a large set of hyperparameters by optimizing the marginal likelihood, overfitting can occur. While a fully Bayesian approach, i.e., marginalizing the hyperparameters, reduces this risk, it generates unacceptable runtimes because the corresponding prediction distribution generally becomes unmanageable analytically. Furthermore, this method does not automatically guarantee certainty or generalizability.

[0004] From DE 10 2017 218 476 A1 a method and a device for determining emissions are known.

[0005] The publication RASMUSSEN, Carl Edward; WILLIAMS, Christophe K.: Gaussian processes for machine learning. Cambridge: .MITPress, 2006 (XVIII). 248 pp. - ISBN 026218253X, provides an introduction to Gaussian processes. Advantages of the invention

[0006] The method with the features of independent claim 1 is a method for training GPs and their sparse approximations by directly optimizing a PAC Bayes bound with respect to its generalization performance. The method is robust and yields significantly better generalization guarantees compared to other, conventional GP approaches. In particular, since no marginalization of the (hyper)parameters is necessary, the new method is more efficient than methods with such marginalization.

[0007] The method therefore does not use a full Bayesian approach to avoid overfitting, but instead minimizes the generalization bounds to ensure that no overfitting occurs. This avoids the computationally intensive marginalization.

[0008] A novel learning objective for general probability (GP) models is revealed, enabling rigorous and quantitatively robust performance guarantees for future predictions. Such rigorous guarantees are the subject of statistical learning theory. However, since classical uniform learning guarantees are ineffective for GPs as non-parametric models, such guarantees cannot be used for learning these models. Conventional optimization objectives instead include (regularized) empirical risk minimization (ERM), maximum likelihood estimation (MLE), or variational inference (VI).

[0009] Improved non-uniform guarantees, however, have been developed within PAC-Bayes theory (PAC = Probably approximately correct). These are specifically tailored to probabilistic methods such as GPs and can provide tight generalization bounds, as in GP classification, probabilistic support vector machine (SVM) methods, linear classifiers, or stochastic neural networks.

[0010] So far, PAC Bayes bounds have mostly been used for the final evaluation of generalization performance, whereas learning by optimizing the PAC Bayes bound has hardly been investigated.

[0011] The present disclosure uses PAC-Bayes bounds for training GPs, for example, in the case of regression, classification, or other types of supervised learning. In particular, the learning of complete and sparse GP predictors Q by directly minimizing an upper PAC-Bayes bound B(Q) to the true future risk R(Q) of the predictor is proposed as a fundamental procedure for ensuring good generalization.

[0012] The Kullback-Leibler divergence (KL divergence) KL(Q||P) in the PAC-Bayes theorem can be analytically evaluated for many pairs of GPs P, Q if they share the same hyperparameters. This applies particularly to common sparse GP variants such as Deterministic Training Conditional (DTC), Fully Independent Training Conditional (FITC), and Variational Free Energy (VFE). These can therefore be used for the new PAC-Bayes learning algorithm, combining advantages in the computation of sparse GPs with theoretical guarantees. In the PAC-Bayes bound, only some of the different possible types of parameters (hyperparameters, inducing points, observation noise, freeform parameters) lead to an increase in the value. According to the revelation, GP learning is based directly on the inverse binary KL divergence and not on previously used looser bounds such as those arising from the Pinsker inequality.

[0013] As mentioned above, the GP learning method can be used, for example, in regression problems, whereas PAC Bayesian bounds have so far only been used in classification cases. However, because a PAC Bayesian bound for regression with a potentially unbounded loss function requires a sub-Gaussian assumption regarding the (unknown) data distribution, a generic bounded loss function is used for the regression in order to remain distribution-free, as in regular PAC theory.

[0014] The learning objective demonstrates robust optimization behavior and scales to larger datasets similarly to other GP methods. In practical applications, the disclosed method showed better risk constraints, in many cases by a factor of two, than conventional methods, and improved guarantees with an increasing number of inducing points.

[0015] The strength of GPs lies in their ability to adapt to different data sizes due to their non-parametric nature. Disclosure of the invention

[0016] In general PAC Bayesian theory and standard supervised learning, a set S with N training data (x) is used. i , y i ) ∈ X × Y(i = 1, ..., N) is used to search in a hypothesis space H⊆YX to train a subset of the space of functions X → Y. Learning algorithms are permitted that define a distribution Q over the hypotheses. h∈H instead of a single hypothesis h, a bounded loss function ℓ: Y × Y → [0,1] is assumed to be given, which is normalized without loss of generality to the interval [0,1]. ℓ(y ∗ ,ŷ) measures how good the prediction ŷ = h(x ∗) an actual output quantity y ∗ to the input variable x ∗ approximates. The empirical risk R S (h) of a hypothesis h is then defined as the mean training loss RS(h):=1N∑i=1Nl(yi,h(xi)). As in conventional PAC theory, an (unknown) underlying distribution µ = µ(x, y) on the set X × Y of data is assumed, and the (real) risk is defined as R(h): = ∫dµ(x, y)ℓ(y, h(x)). To quantify the performance of stochastic learning algorithms that output a distribution Q over hypotheses, the empirical and the real risk are defined as follows: RS(Q):=Eh∼Q[RS(h)]=1N∑i=1NEh∼Q[l(yi,h(xi))] R(Q):=Eh∼Q[R(h)]=E(x*,y*)∼μEh∼Q[l(y*,h(x*))]

[0017] The equations above express the average losses, also called Gibbs risks, for the training and real distributions, respectively, whereby the hypothesis h is drawn from the distribution Q before the prediction.

[0018] The following discussion focuses on the regression case, i.e., where Y ⊆ ℝ is a set of real numbers. A loss function for this case is, for example, l(y*,y^):=1y^∉[r_(y*),r+(y*)] where the functions r ± Specify an interval outside of which a prediction ŷ is considered insufficient. Similar to ε-support vector regression, one can, for example, r±(y*):=y*±ε use with a desired accuracy target ε > 0 specified before training. In any case, the expected values ​​over h ~ Q in equations A and B reduce to one-dimensional integrals, since h(x) ∗ ) at each x ∗ is a real-valued random variable.

[0019] Instead of the stochastic predictor h(x) ∗ ) with h ~ Q is sometimes the deterministic Bayes predictor Eh∼Q[h(x*)] of interest. In GP regression, this is simply equal to the predicted mean m̂(x) ∗ ) at the input value x ∗ The corresponding Bayes risk is defined as RBay(Q):=E(x*,y*)∼μ[l(y*,Eh∼Q[h(x*)])]. While PAC Bayes theorems do not directly provide a bound for R Bay (Q) but only for R(Q), it is easy to see that R Bay (Q) ≤ 2R(Q) if ℓ(y ∗ ,ŷ) quasi-convex in ŷ and the distribution of ŷ = h(x ∗ ) is symmetrical about its mean (e.g., Gaussian distribution). An upper bound B(Q) for R(Q) under 12 This therefore implies a non-trivial bound for R. Bay (Q) ≤ 2B(Q) < 1.

[0020] The following section describes how to train a GP Q by minimizing a suitable risk bound. Due to the probabilistic nature of GPs, generalization bounds for stochastic predictors are used, which often provide better guarantees than generalization bounds for deterministic predictors. Here, PAC-Bayesian bounds are employed.

[0021] The PAC-Bayes theorem states: For every loss function ℓ with values ​​in the interval [0,1], for every distribution µ, for all N ∈ ℕ, for every distribution P over a set H of hypotheses and for each δ ∈ (0,1] the following is true with a probability of at least 1 - δ for the set of training data S ~ µ N to: ∀Q: R(Q)≤kl−1(RS(Q),KL(Q‖P)+ln2NδN).

[0022] This theorem provides a probabilistic upper bound (generalization guarantee) for the true risk R(Q) of a stochastic predictor Q as a function of its empirical risk R. S (Q) on a training dataset S. It requires that a distribution P in the hypothesis space H is determined before the set of training data S is used, and applies to the true risk R(Q) of each distribution Q on H In the following, according to conventional nomenclature, P will also be called the prior distribution (i.e., prior distribution) and Q the posterior distribution (i.e., later distribution) in the PAC Bayesian context. The bound contains a term that can be interpreted as the complexity of the hypothesis distribution Q, namely the KL divergence (Kullback-Leibler divergence). KL(Q‖P):=∫dh Q(h) In Q(h)P(h), with values ​​in [0, +∞]. The bound also includes the binary KL divergence. kl(q‖p):=q ln qp+(1−q)ln1−q1−p, defined for q, p ∈ [0,1], more precisely its (upper) inverse kl -1 Regarding the second argument (for q ∈ [0,1], ε ∈ [0,∞]): kl−1(q,ε):=max{p∈[0,1]:kl(q‖p)≤ε}, which is equal to the uniquely determined p ∈ [q, 1] which satisfies kl(q||p) = ε.

[0023] The right-hand side of equation D can be bounded above by RS(Q)+(KL(Q‖P)+ln2Nδ) / (2N), which, however, can become greater than 1 and thus represent a trivial bound. Furthermore, the complete PAC Bayes theorem simultaneously yields a lower bound for R(Q), which is not relevant since the aim here is to minimize the upper risk bound. Although further refinements of the bound are possible, these are only marginal; therefore, the parameter-free bound from equation C is still considered here.

[0024] If a countable family P θ , parameterized by θ ∈ Θ, used for GP learning with hyperparameters, the above analysis can be generalized by using a probability distribution p θ is determined via Θ and the prior distribution is defined as P:=∑θpθPθ. If 0 is a finite set, then the uniform distribution pθ=1|Θ| a canonical election. One uses the fact that KL(Q‖P)≤KL(Q‖Pθ)+ln1pθ If for every θ ∈ Θ, then the PAC-Bayes theorem yields that with a probability of at least 1 - δ over S ~ µ N The following applies: ∀θ∈Θ ∀Q:R(Q)≤kl−1(RS(Q),KL(Q‖Pθ)+ln1pθ+ln2NδN)=:B(Q).

[0025] The limit from equation E applies simultaneously to all P θ and all Q. Therefore, optimization can be performed over both θ and Q to obtain the best generalization guarantee with a confidence level of at least 1 - δ. B(Q) is used here for training.

[0026] Exemplary applications of the invention include, for example, the modeling of vehicle emissions during real-world driving (RDE emissions). This allows for reliable guarantees regarding emissions, meaning it can be ensured that the emissions are highly likely not to exceed a limit value G.

[0027] Another application is, for example, the assessment of autonomous vehicles. Instead of emissions, a quantitative measure of the hazard state is evaluated, which must not exceed a limit. This indicates, for example, the condition of individual vehicle systems; for instance, a minimum distance to the edge of the road (approximately 0.1 meters) or a minimum distance to every other vehicle (approximately 0.2 meters) must not be undercut.

[0028] Another application is, for example, predicting the state of charge of a battery or its remaining operating time based on historical data, such as usage since the last charging cycle, or environmental data, such as temperature, load or charging current.

[0029] Another application is, for example, in the production of windshield wipers, where a predictor Q is trained with a large amount of data from a production line. This allows for highly reliable predictions of how stable the production line will operate in the next time period (e.g., in the next 30 minutes).

[0030] Therefore, one aspect of the invention provides a method for determining the emission values ​​of a motor vehicle at least partially powered by an internal combustion engine during real-world driving (Real Driving Emissions (RDE)). This involves providing trajectories that characterize the vehicle's driving cycles. These trajectories can be physically recorded using sensors or computer-generated. Crucially, for the reliable determination of the emission values, a Gaussian process is provided, which has been trained using the described learning method. This allows for guarantees regarding the determined emission values.

[0031] For this purpose, the respective emission values ​​of the vehicle are determined using this GP when driving along these provided trajectories, and the emission values ​​in practical driving operation are determined based on these determined emission values. This means that the provided trajectories are preferably a statistically adequate sample of the trajectories actually possible in practical driving operation, so that the emission values ​​in practical driving operation are determined with particular reliability.

[0032] In another aspect, a method for determining the reliability of a technical device is provided. This involves supplying operating cycles of the device, i.e., temporal sequences of its operating states. Reliability can, for example, be a measure that characterizes the fault tolerance of the device or a measure that characterizes its operational safety. Crucially, for the reliable determination of reliability, a Gaussian process is provided, which has been trained using the described learning method. This Gaussian process is configured to determine, based on the trajectories it is given, a measure that characterizes the reliability of the technical device when traversing each of these trajectories.Using this GP (General Process), these parameters will be determined when traversing the provided trajectories, and the reliability will be calculated based on these determined parameters. Thanks to the training method used, reliable statements about the reliability determined in this way are possible.

[0033] In a further aspect, a method for operating the technical device can then be provided in which at least one of the provided trajectories is a trajectory actually traversed by the technical device, and in which the technical device is controlled accordingly depending on the determined reliability.

[0034] If it is then determined that the calculated reliability indicates that a malfunction of the technical device is to be feared, for example because a calculated numerical value characterizing the reliability leaves a predefinable range characterized as "safe", the technical device can be switched to a safe operating mode. Brief description of the drawings

[0035] Exemplary embodiments of the invention are shown in the drawing and explained in more detail in the following description. It shows: Fig. 1 the sequence 100 of a method for training a Gaussian process according to the present invention. Embodiments of the invention

[0036] As explained above, some examples of the loss function in regression (δ = 10) are -2 ; Y = ℝ): l(y,y^,x)=I|y−y^|>ε,l(y,y^,x)=min{((y−y^) / ε)2,1}; l(y,y^,x)=1−exp[−((y−y^) / ε)2] for some selected ε > 0; and l(y,y^,x)=Iy^∉[r−(x,y),r+(x,y)] for some selected functions r_(x, y), r + (x, y): X × Y → ℝ

[0037] In classification, an exemplary loss function is used. l(y,y^,x)=Iy≠sign[y^] (in the binary case y ∈ {-i, +1}).

[0038] It is advantageous if ∫v=−∞+∞N(v|a,b)l(y,v,x)dv and its derivatives with respect to the mean a and the variance b of a univariate Gaussian distribution N(v|a,b) The loss function can be calculated or effectively approximated over v∈ℝ, e.g., analytically or with sufficient numerical accuracy (for all fixed x,y). This can be achieved for all the loss functions mentioned above, partly using the error function.

[0039] According to Fig. 1. When modeling with GPs, a prior distribution is used. P(f)=GP(f|m(x),K(x,x')) The distribution is specified by a positive definite kernel K : X × X → ℝ and a mean function m : X → ℝ on the input set X. In conventional full GP regression, the learned distribution Q is then chosen as the Bayesian posterior distribution, assuming that the training output data yN:=(yi)i=1N∈ℝN faulty versions of f N = (f(x1), ...,f(x N )) are with the independently and identically distributed Gaussian probability yN|fN∼N(yN|fN,σn21). With this assumption, Q is again a GP: Q(f)=GP(f|m(x)+kN(x)(KNN+σn21)−1(yN−mN), K(x,x')−kN(x)(KNN+σn21)−1kN(x')T), where KNN=(K(xi,xj))i,j=1N,kN(x)=(K(x,x1),...,K(x,xN)), and m N = (m(x1), ..., m(x N )). Equation F is used to make (stochastic) predictions for f(x ∗ ) for new input values ​​x ∗ to make ∈ X.

[0040] In the context of the PAC-Bayes bound, it is important that for every complete GP with prior distribution P and corresponding posterior distribution Q according to equation F, the KL divergence KL(Q||P), which appears in the theorem above and in equation E, can be determined on finite-dimensional (N-dimensional) matrices. This makes it possible to effectively determine the PAC-Bayes bound and, based on this, to train GPs by optimizing it. More precisely, it can easily be shown that such P and Q give the same conditional distribution P(f|f). N ) = Q(f|lf N ) have, because a direct calculation yields P(f|fN)=GP(f|m(x)+kN(x)KNN−1(fN−mN),K(x,x')−kN(x)KNN−1kN(x')T=Q(f|fN), which is independent of y N or σ n is, in contrast to Q(f) from equation F. Therefore, the following holds: KL(Q‖P)=KL(Q(fN)Q(f|fN)‖P(fN)P(f|fN))=KL(Q(fN)‖P(fN)) =12ln det[KNN+σn21]−N2ln σn2−12tr[KNN(KNN+σn21)−1]+12(yN−mN)T(KNN+σn21)−1KNN(KNN+σn21)−1(yN−mN), in the last step the known formula for the KL divergence between the normal distributions P(fN)=N(fN|mN,KNN) and Q(fN)=N(fN|mN+KNN(KNN+σn21)−1(yN−mN),KNN−KNN(KNN+σn21)−1KNN) is used and simple simplifications are carried out.

[0041] One goal of training a complete GP is to select "good" values ​​for the hyperparameters θ, which represent a family of prior distributions. Pθ=GP(f|mθ(x),Kθ(x,x')) parameterize, and for the noise level σ n These values ​​are then used to make predictions with the corresponding posterior distribution Q. θ,σn to meet the equation F. The quadratic exponential kernel on X=ℝd,Kθ(x,x')=σs2exp[−12∑i=1d(xi−x'i)2li2], can be used here, whereby σs2 the signal variance is and l i The length scales are, and the mean function is set to zero. The hyperparameters are θ≡(l12,…,ld2,σs2) (SE-ARD core), or θ≡(l2,σs2) if all length scales l1 = ... = l d ≡ l is assumed to be equal (non-ARD).

[0042] The procedure now consists of specifying the parameters θ and σ. n by minimizing the upper bound B(Q θ,σn ) from equation E, and thus the GP predictor Q θ,σn to select with the best generalization performance guarantee, which is guaranteed by the PAC-Bayesian bound. Here, σ n no hyperparameter, since the prior distribution P θ not from σ n depends, and σ n It therefore does not contribute to the increase expression In|Θ|; the parameter σ n is rather a free parameter in the posterior distribution Q θ,σn. Both terms R s (Q θ,σn ) and KL(Qθ,σn‖Pθ) (from equation H), as well as their derivatives, can be efficiently calculated, so that a gradient-based optimization to minimize the right-hand side B(Q) θ,σn ) of equation F over θ and σ n can be used.

[0043] The trained hyperparameters θ can be selected from a discrete set Θ, which must be specified independently of the training data S. This is achieved by defining each of the T components of In θ = (In θ1, ..., In θ) T ) after a gradient-based minimization to the nearest point of the lattice-like (G + 1)-element set {−L,−L+2LG,…,+L}, is discretized, so that In 1pθ=ln|Θ|=T ln(G+1) in the optimization goal B(Q o,σnThe SE-ARD kernel comprises T = d + 1 parameter, while the standard SE kernel comprises T = 2 parameters. For example, each component of In Θ can be rounded to two decimal places in the range [-6, +6], i.e., L = 6, G = 1200. Such discretization often has a negligible effect on the predictive accuracy of Q. θ,σn , while coarser discretization (i.e. a smaller |Θ|) often does not significantly improve either the bounds or the optimization.

[0044] Examples of the kernel function, mean function, or discrete set of hyperparameters are as follows, with X = ℝ d with input dimension d ∈ N: - m(x)=0 (standard mean function). - m(x) given by a physical model of the relation x↦y. - kθ(x,x')=σs2exp[−‖x−x'‖22l2] where the hyperparameters θ=(σs2,l2) are and || · || a Euclidean distance (or a Mahalanobis distance) distance in ℝ d designated. - kθ(x,x')=σs2exp[−∑i=1d(xi−xi')22li2] where the hyperparameters θ=(σs2,l12,…,ld2) are. - For the discretization Θ, each component θ can be i on θ can be discretized as follows: ln θi∈{−L,−L+2LG,…,+L} with L>0 and G∈ℕ, e.g. L=6, G=1201, such that |Θ| ≤ 7.1 × (number of components of θ), where |Θ| denotes the size (cardinality) of the discretized set Θ.

[0045] It is recommended that the initial selections be made before the training dataset is known to guarantee consistent learning. However, a portion of the training dataset that will not be used for training can still be considered. The selections made may also depend on S, but then the learned hypothesis may not adhere to the consistent guarantee suggested by the training goal.

[0046] Despite the fact that with a confidence value of 1 - δ the bound in equation E for each P θ from the Prior GP family and for every distribution Q, the upper bound was determined in the preceding explanation only via the parameters θ,σ. n optimized after P θ and the corresponding Q θ,σn from the equation F. The need to effectively calculate KL(Q||P), focusing on the property Q(f|f N) = P(f|f N ) and the Gaussian form of P(f N ) and Q(f N While supporting (sa equations G and H), it also has a limiting effect. However, building on these two requirements, more general pairs P, Q of GPs with effectively computable KL(Q||P) can be considered, allowing the learning method to be applied more broadly, especially to sparse GP methods.

[0047] Instead of the points x1, ...,x N , which - as above - are taken from the training set S, any number M of points Z = {z1,..., z can be taken from the input space M} ⊆ X can be chosen. These are also called inducing points. Furthermore, any Gaussian distribution can be Q(fM)=N(fM|aM,BMM) on function values fM:=(f(z1),...,f(zM)) with each α M ∈ ℝ M and positive semi-finite matrix B MM ∈ ℝ M×M to be chosen. The distribution Q over f Mcan be applied to all input variables from X using the conditional distribution Q (f|f M ) = P(f|f M ) from the prior distribution (as shown). This yields the following predictive GP: Q(f)=GP(f|m(x)+kM(x)KMM−1(aM−mM), K(x,x')−kM(x)KMM−1[KMM−BMM]KMM−1kM(x')T), where KMM:=(K(zi,zi))i,j=1M,kM(x):=(K(x,z1),...,K(x,zM)), and mM:=(m(z1),...,m(zM)), Using the same reasoning as for equations G and H, we obtain: KL(Q‖P)=KL(Q(fM)‖P(fM))=12ln det[BMMKMM−1]+12tr[BMMKMM−1] −M2+12(aM−mM)TKMM−1(aM−mM).

[0048] Thus, the prior distribution P can be expressed in equation E. θ and a corresponding posterior distribution Q θ,{zi},aM,BMM can be effectively optimized by changing the number M and the locations z1, ...,z M the inducing points and the parameters a M and B MMtogether with the hyperparameters θ. The optimization can be organized using the Sherman-Morrison-Woodbury formula such that for each gradient step the time O(NM) 2 + M 3 ) and the storage O(NM + M 2 ) are needed, compared to O(N 3 ) and O(N 2 ) for complete GPs as described above. For M << N, a large saving occurs.

[0049] Some conventional sparse GP procedures are special cases of the above form, in that certain a M and B MM depending on the training dataset S, such that only the inducing points z1, ... , z M and some other parameters, such as σn2, The following are freely selectable: aM=KMMQMM−1KMN(αΛ+σn21)−1yN,BMM=KMMQMM−1KMM, where QMM=KMM+KMN(αΛ+σn21)−1KNM with KMN:=(K(zi,xj))i,j=1M,N,KNM=KMNT, and Λ = diag(λ1, ...,λ N ) is a diagonal N × N matrix whose entries λi=K(xi,xi)−kM(xi)KMM−1kM(xi)T are. Setting α = 1 corresponds to the FITC approximation, whereas α = 0 yields the procedure according to VFE and DTC. Alternatively, one can also linearly interpolate between the two options using α. Another form of sparse GPs, where the latent function values ​​f M are fixed and are not marginalized over them, corresponds to B MM = 0, which, however, leads via equation J to a diverging KL(Q||P) = ∞ and thus to trivial bounds in equations C and E.

[0050] The learning procedure for sparse GPs then follows the same steps as described above: For the hyperparameters θ, which are suitable to be discretized after the optimization of equation E, an increase expression must be used. ln1pθ=ln|Θ| to be applied. It should be noted that θ only contains the prior hyperparameters, but not the parameters z1, ..., z1. M , a M , B MM , σ n , α from equation K, which are also optimized. The number M can also be varied and thus determines the computational resources. Furthermore, the optimizations can be performed either discretely or continuously. When optimizing over positive semi-finite matrices B MM Can the parameterization B MM = LL T with a lower triangular matrix L ∈ ℝ M×M be used.

[0051] Examples of the optimization goal and the optimization procedure include gradient-based optimization of B(Q) with respect to continuous parameters θ, ρ, where θ must nevertheless be specified in a predefined set Θ. Here, the discretization of the obtained values ​​ρ is not necessary. Alternatively, a different selection procedure for θ, ρ can be applied: Each selection yields a valid upper bound for the generalization risk R(Q) (see equation B). The integrals of the loss function and its derivatives from the above examples are useful for R. S (Q). The derivatives of kl can also be used. -1 calculated or evaluated (for gradient-based optimization). Furthermore, relaxed (i.e., larger) targets can also be used instead of B(Q), such as B. Pin (Q).

[0052] Regarding the applications of the invention, when modeling emissions of a vehicle in real-world driving conditions (RDE emissions), reliable guarantees for the emissions, i.e., ensuring that the emissions are highly likely not to exceed a limit value G, can be obtained, for example, under the following parameters: The above procedure can be used with the loss function l(y,y^,x)=I|y−y^|>ε These methods are used to provide strict guarantees for RDE emissions under typical driving cycles. RDE emissions can be validated against a selected threshold value G. This requires, as a first step, a number D of driving cycles typical for the driving cycle distribution (with an accuracy δ). TV, e.g., in the total variation distance), for which RDE validation is to be performed. These driving cycles can be derived from a set of previously recorded driving cycles and / or generated. The method allows RDE validation to proceed without having to measure emissions during actual test drives (which can only be performed to a limited extent for each specific engine configuration). The method benefits from the availability of a large number of typical driving cycles in step 1 and when simulating emissions in step 2.

[0053] The GP is trained, which can include steps 110 to 150 of the procedure, using a training set specific to the engine configuration under test. A GP Q is output, which can output the (relevant) emissions ŷ for each driving cycle x. ε denotes the GP modeling accuracy (loss function). l(y,y^,x)=I|y−y^|>ε), and δ ∈ (0,1] the confidence parameter for GP modeling.

[0054] In step 1, a number D of driving cycles x1, ..., x is now determined. D , e.g. selected from a previously recorded set or generated by a driving cycle generator, which are typical for the RDE driving cycle distribution, with an accuracy δ TV Then a threshold value G is selected, e.g. the target emission value.

[0055] In step 2, which may involve executing step 160 multiple times, for each of the driving cycles x1, ..., x D , the GP uses to reduce emissions ŷ1, ..., ŷ D to predict.

[0056] In step 3, which includes the evaluation and determination of the guarantee, the number d of predicted emissions ŷ1, ..., ŷ can be used to determine D , which exceed the value G-ε together with D, δ, δ TV and a further level of confidence δ CPUsing standard statistical techniques (especially the Clopper-Pearson Lemma), a strict guarantee (upper bound) can be given for the proportion of typical driving cycles that violate the RDE requirements. The guarantee can be highly likely to exist if δ, δ TV , S CP are small.

[0057] When assessing autonomous vehicles, a quantitative measure of the hazard state is evaluated, which must not exceed a limit. This indicates, for example, the condition of individual vehicle systems; for instance, a minimum distance to the edge of the road (approximately 0.1 meters) or a minimum distance to every other vehicle (approximately 0.2 meters) must not be undercut.

[0058] In this approach, compared to the above, the emissions (i.e., the actual emission values ​​y, the predicted emission values ​​ŷ, and the emission threshold G) are replaced by a quantitative measure of hazard, indicating, for example, how close the autonomous driving system (or one of its subsystems) is to a state of failure. Such a measure must be selected taking into account a specific application, such as the minimum distance to the edge of the road that must not fall below a certain threshold in each driving cycle, e.g., 0.1 meters. Or the minimum distance to any other vehicle on the road that must not fall below a certain threshold in each driving cycle, e.g., 0.2 meters.

[0059] Hazard values ​​can be predicted for a number D of selected driving cycles, so that a strict guarantee (up to ±ε, and with a probability ≥ 1 - δ) for the hazard values ​​can be obtained using the GP method. A guarantee can then also be obtained using standard statistical techniques (especially the Clopper-Pearson Lemma).

[0060] The GP is trained, which can include steps 110 to 150 of the procedure, using a training set specific to the autonomous driving system under test. A GP Q is output, which can output the hazard values ​​ŷ for each driving cycle x. ε denotes the GP modeling accuracy (loss function). l(y,y^,x)=I|y−y^|>ε), and δ ∈ (0,1] the confidence parameter for GP modeling.

[0061] In step 1, a number D of driving cycles x1, ..., x is now determined. De.g. selected from a previously recorded set or generated by a driving cycle generator, which are typical driving cycles, with an accuracy δ TV Then a threshold value G is selected, e.g. for the level of danger.

[0062] In step 2, which may involve executing step 160 multiple times, for each of the driving cycles x1, ...,x D , the GP uses to determine the hazard values ​​ŷ1, ..., ŷ D to predict.

[0063] In step 3, which includes the evaluation and determination of the guarantee, the number d of predicted emissions ŷ1, ..., ŷ can be used to determine D , which exceed the value G-ε together with D, δ, δ TV and a further level of confidence δ CPUsing standard statistical techniques (especially the Clopper-Pearson Lemma), a strict guarantee (upper bound) can be given for the proportion of typical driving cycles that exceed the hazard threshold G. The guarantee can be highly likely to exist if δ, δ TV , S CP are small.

[0064] The method can also be used to predict the state of charge of a battery or its remaining operating time, based on historical data, such as usage since the last charging cycle, or environmental data, such as temperature, load or charging current.

[0065] For this purpose, GP Q is trained, which can comprise steps 110 to 150 of the procedure, starting from many (N) measurements of input values ​​x (e.g., battery history, environmental parameters) and corresponding output values ​​y (e.g., state of charge, remaining operating time). The loss function can be... l(y,y^,x)=I|y−y^|>ε, This can be used with an accuracy target ε and the desired confidence parameter δ. This training step is only executed at the beginning. The following steps can be executed as a loop until the end of the battery life.

[0066] In step 1, for a specific application such as a battery-powered vehicle, input values ​​x, e.g. from sensors or a storage device, are read in. Such input values ​​are conventionally received repeatedly during the execution of the process.

[0067] In step 2, the trained GP predicts the output values ​​ŷ from the input values ​​x.

[0068] In step 3, if the predicted ŷ exceed or fall below a previously selected threshold G, a signal (e.g. a warning light) is given to warn the user, the battery load is changed, e.g. by switching off a consumer such as a radio or air conditioner, or switched to or added to an alternative energy source, such as an internal combustion engine in a hybrid vehicle.

[0069] The nature of the GP learning algorithm, which takes place before step 1, ensures that for each prediction ŷ, the algorithms miscalculate by a maximum value of ±ε with a minimum probability of 1 - δ. Thus, a statistical guarantee can be given that the action triggered in step 3 will not be performed if the actual battery status (i.e., the actual output value y) is below G-ε, and conversely, that the action in step 3 will be triggered if the battery status is above G+ε. This ensures that the system operates as intended.

[0070] In windshield wiper production, a predictor Q is trained using a large amount of data from a production line. This allows for highly reliable predictions of how stable the production line will operate in the next time period (e.g., the next 30 minutes). Thus, the GP modeling method described above (which provides a provable, statistical guarantee of prediction performance with new data) can be used to train a GP predictor Q from a large amount of production line data (e.g., from windshield wiper production) to predict how stable the production line will be in an upcoming period (e.g., the next 30 minutes). If the prediction is that the production line will operate stably enough (i.e.,(with a sufficiently large distance ε from the GP loss function, and with a sufficiently large confidence level 1 - δ from the PAC Bayes target), the monitoring personnel can reduce their attention or maintenance efforts and pursue other activities or take a break.

[0071] Since downtime incurs high costs, reducing monitoring efforts is only feasible if there is a strict guarantee of the production line's correct operation. This is achieved through the learning process described above. The asymmetry of the task must be considered here: while reducing monitoring can potentially lead to high one-off costs if the production line fails, continuous monitoring, even when the production line is functioning perfectly, has significantly less devastating consequences.

[0072] In a first step, GP Q is trained, which can comprise steps 110 to 150 of the procedure, starting from many (N) measurements of input values ​​x (e.g., temperature T, and parameters κ dependent on the production machinery) and corresponding output values ​​y (e.g., quality of the produced parts). The loss function can be... l(y,y^,x)=I|y−y^|>ε, This can be used with a target accuracy ε and the desired confidence parameter δ. This training step is performed at the beginning, but can be repeated later in a loop with a larger data set if necessary.

[0073] In step 2, the trained GP is tested at the currently predicted temperature development and for various adjustable parameter settings κ ievaluated to find the optimal (good) parameter settings κ for the planning horizon of the next M slices, such that (i) a low predicted reject rate (GP mean) is achieved, and (ii) the trained GP has a low guaranteed loss (i.e., accurate prediction)

[0074] In step 3, the (human) operator is presented with parameters κ for the next M of discs to be produced. Furthermore, the operator is provided with a confidence estimate regarding the reliability of the GP prediction in achieving the estimated low reject rate over the time window M.

[0075] If the confidence level reaches or exceeds a preset value, a green light, for example, can illuminate. Based on this confidence level, the operator can then decide whether the machine can be left unattended or not.

[0076] In step 4, once the machine is operating as configured and producing new parts, the quality of these new parts can be measured. The measurements y can then be used as additional data to retrain the GP with larger training datasets. This will improve the GP predictions over time, particularly in those parts of the parameter space where the GP initially did not predict accurately or reliably.

[0077] Then either step 1 can follow, otherwise steps 2 and 3 follow again in the loop.

Claims

[1] Computer-implemented method for determining emission values ​​of a motor vehicle powered at least partially by an internal combustion engine in real-world driving conditions (Real Driving Emissions (RDE)), wherein trajectories characterizing driving cycles of the motor vehicle are provided, and wherein a Gaussian process (GP) is provided which determines the emissions of the motor vehicle when driving through the respective trajectories supplied to it, depending on the trajectories supplied to it, and wherein the respective emission values ​​of the motor vehicle when driving through these supplied trajectories are determined by means of this GP, and wherein the emission values ​​in real-world driving conditions are determined depending on these determined emission values. characterized by that Training the GP from a GP parameterization is performed by optimizing a PAC Bayes bound (150) using a training dataset, a prior GP family, and an accuracy target. [2] A computer-implemented method for determining the reliability of a technical device, in particular a motor vehicle assembly, wherein trajectories characterizing the operating cycles of the technical device, in particular driving cycles of the motor vehicle, are provided, and wherein a Gaussian process, GP, is provided which, depending on the trajectories supplied to it, determines a quantity that characterizes the reliability of the technical device, in particular the motor vehicle assembly, when driving through each of these supplied trajectories, and wherein, by means of this GP, this quantity is determined depending on the supplied trajectories. and where the reliability is determined depending on these determined quantities, characterized by that Training the GP from a GP parameterization is performed by optimizing a PAC Bayes bound (150) using a training dataset, a prior GP family, and an accuracy target. [3] Method according to claim 2, wherein the technical device is an assembly of a motor vehicle, and this assembly comprises a system for at least semi-autonomous operation of the motor vehicle and / or a battery for powering an electric motor of the motor vehicle. [4] Method according to claim 2, wherein the technical device is a manufacturing machine. [5] Method for operating a technical device, wherein the reliability of the technical device is determined by means of the method according to one of claims 2 to 4, wherein at least one of the provided trajectories includes a trajectory actually traversed by the technical device, and wherein the technical device is controlled accordingly depending on the determined reliability. [6] Method according to claim 5, wherein the technical device is switched to a safe operating mode if the determined reliability indicates that a malfunction of the technical device is to be feared. [7] Computer-implemented method (100) according to any one of claims 1 to 6, wherein the method (100) further comprises the following steps: Before training the GP, select the accuracy target (110); Before training the GP, select the prior GP family (120) that is parameterized by hyperparameters; Before training the GP, obtain the training dataset (130); Before training the GP, select the GP parameterization for modeling (140); Predictions (160) of the next expected values ​​by the trained GP in response to receiving new input values ​​x. [8] Computer-implemented method (100) according to any one of claims 1 to 7, wherein the accuracy target may include loss function and confidence parameters, wherein the loss function depends on the modeling task, wherein the modeling task is one of classification, regression or another supervised learning task. [9] Computer-implemented method (100) according to any one of claims 1 to 8, wherein the GP parameterization for modeling (140) is a complete GP, free sparse GP, or a special parameterization of a sparse GP such as Deterministic Training Conditional, DTC, Fully Independent Training Conditional, FITC, or Variational Free Energy, VFE. [10] Computer-implemented method (100) according to claim 7, wherein the hyperparameters parameterize the kernel function and the mean value function and the possible values ​​of the hyperparameters are in a predefined list. [11] Computer-implemented method (100) according to any one of claims 1 to 10, wherein values ​​predicted by the GP are one of the following: Emissions of a vehicle during real-world driving (English: "Real Driving Emissions (RDE)") quantitative values ​​of a hazardous state of an autonomous vehicle, values ​​relating to the state of charge of a battery or its remaining operating time, and Values ​​relating to production, and in particular the production of windshield wipers. [12] System comprising a GP whose training is carried out from a GP parameterization by optimizing a PAC Bayes bound (150) using a training data set, a prior GP family, and an accuracy target. [13] Computer program comprising instructions which are set up to execute the method according to any one of claims 1 to 11 when executed on a computer. [14] Machine-readable storage medium on which the computer program according to claim 13 is stored. [15] Device configured to carry out the method according to any one of claims 1 to 11.

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